Joint monitoring of mean and variance using Max-EWMA control chart under lognormal process with application to engine oil data

The control charts are frequently employed in process monitoring to assess the average and variability of a process, assuming a normal distribution. However, it is worth noting that some process distributions tend to exhibit a positively skewed distribution, such as the lognormal distribution. This article proposed a maximum exponential weighted moving average control chart for joint monitoring of mean and variance under a lognormal process. The proposed control chart is evaluated by using the run length profile such as ARL and SDRL. The Monte Carlo simulation is conducted by using the R language to find the run length profile. An application is presented to demonstrate the design of the proposed control chart.


Design of proposed control chart
In this section, we discuss the proposed Max-EWMA control chart for joint monitoring of mean and variance with lognormal distribution.Let X 1 , X 2 , X 3 , . . ., X i ; i = 1, 2, 3, 4, . . .n, follow the lognormal distribution with respective location and scale parameters, that is µ x and σ x with a probability density function

The CDF of lognormal distribution is
To convert the mean and variance of a lognormal distribution to a normal distribution, one can use the properties of logarithms For each data point x from the lognormal distribution, compute Y = ln(x).The resulting values of Y should follow a normal distribution with mean µ y and standard deviation σ y .
The current Shewhart control chart makes use of all the available information in the current sample.On the other hand, the EWMA control chart was designed to provide greater importance to the most recent subgroup, while assigning decreasing weights to previous observations in a geometric manner.The EWMA statistic is (1) x www.nature.com/scientificreports/employed specifically to analyze shifts in the mean of a process.Roberts 23 recommended EWMA statistic for the jth sample of size n for examining mean parameter as given in (3).
Chen et al. 24 introduced the concept of simultaneously monitoring both the mean shift and variance shift using a single control chart, which they referred to as the Max-EWMA control chart.Let Y represent a normally distributed random variable in the context of process production.The mean of Y is given by µ y = µ 0 + aσ 0 and σ 2 y = b 2 σ 2 0 , where µ 0 and σ 2 0 are respective parameters for the mean and variance when the process is in a stable state.The variables a and b represent shifts in mean and variance, respectively.In an in-control process, the values of a and b are 0 and 1, respectively.The utilization of Max-EWMA statistic has been found to exhibit enhanced efficiency in detecting minor shifts.The statistical properties of mean and variance for in-control methods, specifically for the jth sample, follow a normal distribution with a mean of zero and a variance of one. of jth sample, The inverse function of the usual normal distribution is denoted as ∅ −1 .On the other hand, the random variable H(ξ , v) follows a chi-squared distribution with n−1 degrees of freedom.
The two EWMA statistics can be derived by utilizing Eqs. ( 4) and (5).
In the first sample, the values of P 0 and Q 0 are both set to zero.The smoothing constant, denoted as λ, is defined such that it falls within the range of 0-1, inclusive.
In the suggested statistical framework, the sum of all weights is equal to one.The variables U j−1 and V j−1 rep- resent the previous values of the respective quantities.In order to streamline the analysis of mean and variance, we propose employing a single maximum absolute statistic for their joint monitoring of mean and variance, as opposed to analyzing these statistics individually.
A single upper control limit (UCL) is sufficient for the representation of Max-EWMA statistic to regulate the production process.The UCL in Eq. ( 9) draws inspiration from the research conducted by Chen et al. 24 .For further insights into the specific values employed in Eq. (9), readers are encouraged to refer to the work of Chen et al. 24 .
The control constant indicated as L, is determined for achieving the desired average run length (ARL 0 ) in an in-control process.Additionally, the variance is also included in this calculation.

Performance evaluation
The evaluation of the control chart is typically conducted by the analysis of its run length attributes, ARLs and SDRLs.When the process is under control, it is expected that a control chart would possess a large value of ARL, indicating that control chart would infrequently signal any deviations from the established control limits.In contrast, in instances where the process is not under control, a control chart should possess a small ARL, indicating its ability to promptly detect signals indicating an out-of-control state.In the context of control charts, it is generally regarded that the control chart exhibiting a reduced ARL when subjected to a particular shift parameter is deemed to be superior in terms of performance.When the process is operating in a state of control, it is necessary to implement measures to minimize the occurrence of frequent false alarms.To promptly identify any shifts in the out-of-control process, it is imperative that the ARL be minimized.The ARL and SDRL at any shift are denoted by ARL 1 and SDRL 1.A lower value of SDRL serves as an indicator of improved performance of a control chart.Monte Carlo simulations are employed to calculate the ARLs with each run consisting of 30,000 repeats.During the duration of the analysis, it is necessary for the calculated value of the plotting statistic T j to fall under the UCL.The following steps are considered to compute the run length profile: (3)

ARL 0 calculation
Compute ARL 0 by repeating the above steps (1-4) for 30,000 times.Should the target ARL 0 not be achieved, redo the preceding steps by using a different value for L. The selected value of L for a specific ARL 0 will be used in the next step.6. ARL 1 calculation Select the shifted parameters of the lognormal distribution L(α , γ ).Repeat the steps 1 to 4 and compute the value of ARL 1 by repeating the process 30,000 times.
The shift occurs when the value of the plotted statistic falls outside the UCL.The impact of variations in lognormal parameters on the process mean and process standard deviation is taken into consideration.In this context, the shape parameters undergo variations at the values of 0.5, 1, 1.25, 1.5, 1.75, 1.9, 2, 2.16, 2.25, 2.5, 2.75, 3.00, 4.00 and 5.00.While the scale parameters experience changes at the values of 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.649, 1.75, 2, 2.25, 2.5, 2.75, 3.00, 4.00, and 5.00.The process at point L(1.65, 2.16) is determined to be incontrol, with the associated normal process following a distribution of N(0, 1).The value of the in-control ARL 0 is set to 370.With two different settings of smoothing parameter = 0.3 and 0.2.The corresponding values of L are 3.345 and 3.252, respectively.The ARLs and SDRLs for the proposed Max-EWMA control chart, considering different changes in shape and scale parameters, are presented in Tables 1 and 2, respectively.The bold values of ARLs and SDRLs are presented in Tables 1 and 2 for the fixed value of the scale parameter and various values of the shape parameter.The parametric settings to calculate the ARLs and SDRLs are presented in Table 3.

Main findings
1.By holding the scale constant α = 1.649, and varying the shape parameter, a diverse array of outcomes can be attained.The alterations in mean and standard deviation correspond directly to the magnitude and direction of the shape parameter adjustment.Specifically, for shape parameter values exceeding 2.16, even with an ARL 0 = 370 indicating control, the normal process can manifest fluctuations in both its mean and standard deviation.For α = 1.64, we observed the change in transformed mean and standard deviation with the change in shape parameter.Because of this modification, the mean and standard deviation will move together, resulting in lower values of ARL 1 .For instance, when considering increased sample sizes, the ARL 1 falls significantly.Specifically, when the sample size is 3, the ARL 1 decreases from 217 to 3.17.Similarly, for a sample size of 5, the ARL 1 decreases from 188 to 2.27.Furthermore, when the sample size is 10, the ARL 1 decreases from 138 to 1.58.In the case of a downward shift ( γ < 2.16).The changes in the ARL 1 decreased trend from 214.92 to 2.83 for a sample size of 3, from 92.93 to 2.00 for sample size 5 and from 30.96 to 0.17 for sample size 10. 2. When altering the scale parameter while maintaining the shape parameter constant, a downward shift is noticeable.This adjustment leads to a more significant displacement in the corresponding standard parameters compared to the upward displacement.The magnitude of the shift in the Lognormal scale parameter is directly proportional to the magnitude of the change in a normal parameter.For instance, when α = 5.00, the shift in the mean and standard deviation is determined to be 1.5238 and 0.1713, respectively.3. When holding the scale and shape parameters constant and manipulating the sample size, it is seen that the ARL 1 and the SDRL 1 drop as the sample size increases.This implies that if the producer is able to handle a big sample size, it is possible to promptly identify an out-of-control process by raising the sample size while maintaining the same scale and shape parameter.For Example in   19 where they determined that the probability volume integral signature adheres to a lognormal distribution.The concept of distribution refers to the way in which resources, goods, or services are allocated.There exists a specific type of reference oil, namely Ref Oil 434, which possesses certain quality characteristics that require attention.The experiment was conducted.In order to establish a reference for oils, a total of 50 samples, each consisting of 10 units, were gathered and subsequently employed for analysis.The data about the Ref Oil 434 are shown in Table 4.The reader may consult Huang et al. 19 for further details on the dataset.In the case of Ref Oil 434, as depicted in Figs. 1, 2, and 3, it is observed that there are instances of out-of-control samples occurring at the 21st position for sample sizes of n = 3 and 19th position for sample size of n = 5.Additionally, sample 17 is found to be outside the control limits with a sample size of n = 7, as indicated by the proposed control chart for λ = 0.2.Hence, it may be inferred that as the sample size increases, the efficiency of detecting instances of being out of control also increases.When λ = 0.2, as illustrated in Figs. 4, 5, and 6, it is evident that there are occurrences of out-of-control samples at the 24th position for a sample size of n = 3 and at the 19th position for a sample size of n = 5.Furthermore, it has been seen that sample 17, with a sample size of n = 7, falls outside the control limits as depicted by the suggested control chart.Therefore, it can be deduced that as the size of the sample increases, the ability to detect instances of being out of control likewise improves.

Conclusion
The establishment of Max-EWMA control charts for concurrently tracking the lognormal distribution's mean and standard deviation is covered in this paper.According to the simulation experiments, the Max-EWMA charts perform well in cases where the underlying lognormal distribution is more skewed.An increase in the lognormal shape parameter (λ > 2.16) results in a little change in both the mean and standard deviation.As the value of λ grows, there is a decrease observed in both the mean and standard deviation shifts.A decrease in the lognormal shape parameter (λ < 2.16) is associated with reductions in both the mean and standard deviation.As the value of λ declines, there is an increase in the shift observed in the standard deviation of the normal parameter.This increase in shift leads to a reduction in the ARL 1 values from 217 to 3.17.The inverse response function is employed for the purpose of transformation.

Table 2
When the scale and shape parameters (2, 2.25) are kept constant and the smoothing constant is varied, it is observed that decreasing the smoothing constant leads to a drop in both the ARL 1 and the SDRL 1 values.

Table 3 .
parameters for the proposed control chart.